The average of three integers is 13. When a natural number $$ n $$ is included, the average of these four integers remains an odd integer. The minimum possible value of $$ n $$ is:

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To solve this problem:

1. Let the three integers be $$ I_1, I_2, I_3 $$. Their average is given as 13, so their sum is:  
   $$
   I_1 + I_2 + I_3 = 13 	imes 3 = 39
   $$

2. When a fourth integer $$ I_4 $$ (natural number $$ n $$) is added, the new average becomes:  
   $$
   	ext{New Average} = \frac{39 + I_4}{4}
   $$

3. For the new average to remain an **odd integer**, the numerator $$ 39 + I_4 $$ must also be odd (since the denominator is even). Thus:  
   $$
   39 + I_4 = 	ext{odd} \implies I_4 = 	ext{even}
   $$

4. To find the **minimum value of $$ I_4 $$** such that the average remains odd:  
   - The new average can be expressed as $$ 13 + x $$, where $$ x = \pm 1, \pm 2, ...$$.  
   - Solving for $$ I_4 = 13 + 8x $$, we find that $$ x = -1 $$ gives the smallest natural number:  
     $$
     I_4 = 13 - 8(1) = 5
     $$

Thus, the minimum possible value of $$ n (I_4) $$ is **5**.

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